L(s) = 1 | + (−1.37 + 2.38i)2-s + (−0.745 + 1.29i)3-s + (−2.80 − 4.85i)4-s + (−0.5 + 0.866i)5-s + (−2.05 − 3.56i)6-s − 2.84·7-s + 9.94·8-s + (0.387 + 0.670i)9-s + (−1.37 − 2.38i)10-s − 0.864·11-s + 8.36·12-s + (−0.321 − 0.557i)13-s + (3.92 − 6.80i)14-s + (−0.745 − 1.29i)15-s + (−8.11 + 14.0i)16-s + (−1.87 + 3.24i)17-s + ⋯ |
L(s) = 1 | + (−0.975 + 1.68i)2-s + (−0.430 + 0.745i)3-s + (−1.40 − 2.42i)4-s + (−0.223 + 0.387i)5-s + (−0.839 − 1.45i)6-s − 1.07·7-s + 3.51·8-s + (0.129 + 0.223i)9-s + (−0.436 − 0.755i)10-s − 0.260·11-s + 2.41·12-s + (−0.0892 − 0.154i)13-s + (1.04 − 1.81i)14-s + (−0.192 − 0.333i)15-s + (−2.02 + 3.51i)16-s + (−0.453 + 0.785i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.619 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.139971 - 0.288567i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.139971 - 0.288567i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (3.36 - 2.77i)T \) |
good | 2 | \( 1 + (1.37 - 2.38i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.745 - 1.29i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 2.84T + 7T^{2} \) |
| 11 | \( 1 + 0.864T + 11T^{2} \) |
| 13 | \( 1 + (0.321 + 0.557i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.87 - 3.24i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (0.208 + 0.361i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.85 - 8.40i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.93T + 31T^{2} \) |
| 37 | \( 1 - 6.36T + 37T^{2} \) |
| 41 | \( 1 + (-2.00 + 3.47i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.02 + 1.78i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.97 - 3.42i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.49 - 9.51i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.22 - 2.13i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.16 + 5.48i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.26 - 2.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.891 + 1.54i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.56 + 6.17i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.912 - 1.58i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.43T + 83T^{2} \) |
| 89 | \( 1 + (2.22 + 3.85i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.42 + 9.39i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.22332619573006294703783937380, −14.13032776629690190991523127697, −12.88489749321740033932764716291, −10.64910981063371011645180329000, −10.24653436607945490662953401297, −9.117115397968429684707187259800, −7.941504686739693209638942057637, −6.72217172471589758671393641226, −5.82261536358874283246359101200, −4.40739101166759622335440238493,
0.55146671449796190213619435113, 2.60848274395953931291229738353, 4.26193114455139473223999618631, 6.72366172498670331588153497256, 8.050315054870472181338719935829, 9.275499103938213935861210231558, 10.02522220406436784616903418514, 11.38045374129735384443647220725, 12.05084360794703229461209744787, 12.98074358310946157188842845502